What are the domain and range of the function $y = 2 ext{cos}^{-1}(2x) + 2 ext{sin}^{-1}(2y)$? - HSC - SSCE Mathematics Extension 1 - Question 4 - 2024 - Paper 1
Question 4
What are the domain and range of the function $y = 2 ext{cos}^{-1}(2x) + 2 ext{sin}^{-1}(2y)$?
Worked Solution & Example Answer:What are the domain and range of the function $y = 2 ext{cos}^{-1}(2x) + 2 ext{sin}^{-1}(2y)$? - HSC - SSCE Mathematics Extension 1 - Question 4 - 2024 - Paper 1
Step 1
Domain: Determine the valid input values for $x$
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Answer
Given the inverse functions involved, the argument of extcos−1 and extsin−1 must fall within the ranges of
−1≤2x≤1 for extcos−1 and −1≤2y≤1 for extsin−1.
Setting these inequalities gives us:
From −1≤2x≤1:
Dividing everything by 2 results in:
−0.5≤x≤0.5.
Similarly, for y:
−1≤2y≤1 gives:
−0.5≤y≤0.5.
Thus, considering both variables, the domain can be specified as:
[−0.5,0.5].
Step 2
Range: Determine the output values of the function
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Answer
The range of the function is influenced by the inverse trigonometric functions:
For extcos−1(x), the output is limited to [0,pi]. Since we have 2extcos−1(2x), this stretches the output to [0,2pi].
For extsin−1(y), the output is [−fracπ2,fracπ2], and thus, 2extsin−1(2y) alters the range to [−π,π].
Considering the combination of these outputs, we can conclude that the range of the overall function is:
{π}.
